Running Head: G*Power 3
G*Power 3: A flexible statistical power analysis program
for the social, behavioral, and biomedical sciences
(in press). Behavior Research Methods.
Franz Faul
Christian-Albrechts-Universität Kiel
Kiel, Germany
Edgar Erdfelder
Universität Mannheim
Mannheim, Germany
Albert-Georg Lang and Axel Buchner
Heinrich-Heine-Universität Düsseldorf
Düsseldorf, Germany
Please send correspondence to:
Prof. Dr. Edgar Erdfelder
Lehrstuhl für Psychologie III
Universität Mannheim
Schloss Ehrenhof Ost 255
D-68131 Mannheim, Germany
Email: [email protected]
Phone +49 621 / 181 – 2146
Fax: + 49 621 / 181 - 3997
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Abstract
G*Power (Erdfelder, Faul, & Buchner, Behavior Research Methods, Instruments, & Computers,
1996) was designed as a general stand-alone power analysis program for statistical tests commonly
used in social and behavioral research. G*Power 3 is a major extension of, and improvement over,
the previous versions. It runs on widely used computer platforms (Windows XP, Windows Vista,
Mac OS X 10.4) and covers many different statistical tests of the t-, F-, and !2-test families. In
addition, it includes power analyses for z tests and some exact tests. G*Power 3 provides improved
effect size calculators and graphic options, it supports both a distribution-based and a design-based
input mode, and it offers all types of power analyses users might be interested in. Like its
predecessors, G*Power 3 is free.
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G*Power 3: A flexible statistical power analysis program
for the social, behavioral, and biomedical sciences
Statistics textbooks in the social, behavioral, and biomedical sciences typically stress the
importance of power analyses. By definition, the power of a statistical test is the probability of
rejecting its null hypothesis given that it is in fact false. Obviously, significance tests lacking
statistical power are of limited use because they cannot reliably discriminate between the null
hypothesis (H0) and the alternative hypothesis (H1) of interest. However, although power analyses
are indispensable for rational statistical decisions, it took until the late 1980s until power charts
(e.g., Scheffé, 1959) and power tables (e.g., Cohen, 1988) were supplemented by more efficient,
precise, and easy-to-use power analysis programs for personal computers (Goldstein, 1989).
G*Power 2 (Erdfelder, Faul, & Buchner, 1996) can be seen as a second-generation power analysis
program designed as a stand-alone application to handle several types of statistical tests commonly
used in social and behavioral research. In the past ten years, this program has been found useful
not only in the social and behavioral sciences but also in many other disciplines that routinely
apply statistical tests, for example, biology (Baeza & Stotz, 2003), genetics (Akkad et al., 2006),
ecology (Sheppard, 1999), forest- and wildlife research (Mellina, Hinch, Donaldson, & Pearson,
2005), the geosciences (Busbey, 1999), pharmacology (Quednow et al., 2004), and medical
research (Gleissner, Clusmann, Sassen, Elger, & Helmstaedter, 2006). G*Power 2 was evaluated
positively in the reviews we are aware of (Kornbrot, 1997; Ortseifen, Bruckner, Burke, & Kieser,
1997; Thomas & Krebs, 1997), and it has been used in several power tutorials (e.g., Buchner,
Erdfelder, & Faul, 1996, 1997; Erdfelder, Buchner, Faul & Brandt, 2004; Levin, 1997; Sheppard,
1999) as well as in statistics textbooks (e.g., Field, 2005; Keppel & Wickens, 2004; Myers & Well,
2003; Rasch, Friese, Hofmann, & Naumann, 2006a, 2006b). Nevertheless, the user feedback that
we received converged with our own experience in showing some limitations and weaknesses of
G*Power 2 that required a major extension and revision.
The present article describes G*Power 3, a program that was designed to address the
problems of G*Power 2. We will first outline the major improvements in G*Power 3 (Section 1)
before we discuss the types of power analyses covered by this program (Section 2). Next, we
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describe program handling (Section 3) and the types of statistical tests to which it can be applied
(Section 4). The fifth section is devoted to the statistical algorithms of G*Power 3 and their
accuracy. Finally, program availability and some internet resources supporting users of G*Power 3
are described in Section 6.
1) Improvements in G*Power 3 compared to G*Power 2
G*Power 3 is an improvement over G*Power 2 in five major aspects. First, whereas G*Power 2
requires the DOS and Mac-OS 7-9 operating systems that were common in the 1990s but are now
outdated, G*Power 3 runs on the currently most widely used personal computer platforms, that is,
Windows XP, Windows Vista, and Mac OS X 10.4. The Windows and the Mac versions of the
program are essentially equivalent. They use the same computational routines and share very
similar user interfaces. For this reason, we will not differentiate between both versions in the
sequel; users simply have to make sure to download the version appropriate for their operating
system.
Second, whereas G*Power 2 is limited to three types of power analyses, G*Power 3 supports five
different ways to assess statistical power. In addition to a priori analyses, post hoc analyses, and
compromise power analyses that were already covered by G*Power 2, the new program also offers
sensitivity analyses and criterion analyses.
Third, G*Power 3 provides dedicated power analysis options for a variety of frequentlyused t tests, F tests, z tests, !2 tests, and exact tests, rather than just the standard tests covered by
G*Power 2. The tests captured by G*Power 3 are described in Section 3 along with their effect
size parameters. Importantly, users are not limited to these tests because G*Power 3 also offers
power analyses for generic t-, F-, z-, !2, and binomial tests for which the noncentrality parameter
of the distribution under H1 may be entered directly. In this way, users are provided with a flexible
tool for computing the power of basically any statistical test that uses t-, F-, z-, !2, or binomial
reference distributions.
Fourth, statistical tests can be specified in G*Power 3 using two different approaches, the
distribution-based approach and the design-based approach. In the distribution-based approach,
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users select (a) the family of the test statistic (i.e., t-, F-, z-, !2, or exact test) and (b) the particular
test within this family. This is the way in which power analyses were specified in G*Power 2.
Additionally, a separate menu in G*Power 3 provides access to power analyses via the designbased approach: Users select (a) the parameter class the statistical test refers to (i.e., correlations,
means, proportions, regression coefficients, variances) and (b) the design of the study (e.g.,
number of groups, independent vs. dependent samples, etc.). Based on the feedback we received to
G*Power 2, we expect that some users might find the design-based input mode more intuitive and
easier to use.
Fifth, G*Power 3 supports users with enhanced graphics features. The details of these
features will be outlined in Section 3, along with a description of program handling.
2) Types of Statistical Power Analyses
The power (1-") of a statistical test is the complement of ", which denotes the type-2 or beta error
probability of falsely retaining an incorrect H0. Statistical power depends on three classes of
parameters: (1) the significance level (or, synonymously, the type-1 error probability) # of the test,
(2) the size(s) of the sample(s) used for the test, and (3) an effect size parameter defining H1 and
thus indexing the degree of deviation from H0 in the underlying population. Depending on the
available resources, the actual phase of the research process, and the specific research question,
five different types of power analysis can be reasonable (cf. Erdfelder et al., 2004; Erdfelder, Faul,
& Buchner, 2005). We describe these methods and their uses in turn.
1) In a priori power analyses (Cohen, 1988), the sample size N is computed as a function
of the required power level (1-"), the pre-specified significance level #, and the population effect
size to be detected with probability (1-"). A priori analyses provide an efficient method of
controlling statistical power before a study is actually conducted (e.g., Bredenkamp, 1969; Hager,
2006) and can be recommended whenever resources such as time and money required for data
collection are not critical.
2) In contrast, post hoc power analyses (Cohen, 1988) often make sense after a study has
already been conducted. In post hoc analyses, the power (1-") is computed as a function of #, the
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population effect size parameter, and the sample size(s) used in a study. It thus becomes possible
to assess whether a published statistical test in fact had a fair chance to reject an incorrect H0.
Importantly, post-hoc analyses, like a priori analyses, require an H1 effect size specification for the
underlying population. They should not be confused with so-called retrospective power analyses in
which the effect size is estimated from sample data and used to calculate the “observed power”, a
sample estimate of the true power1. Retrospective power analyses are based on the highly
questionable assumption that the sample effect size is essentially identical to the effect size in the
population from which it was drawn (Zumbo & Hubley, 1998). Obviously, this assumption is
likely to be false, the more so the smaller the sample. In addition, sample effect sizes are typically
biased estimates of their population counterparts (Richardson, 1996). For these reasons, we agree
with other critics of retrospective power analyses (e.g., Gerard, Smith & Weerakkody, 1998;
Hoenig & Heisey, 2001; Kromrey & Hogarty, 2000; Lenth, 2001; Steidl, Hayes, & Schauber,
1997). Rather than using retrospective power analyses, researchers should specify population
effect sizes on a priori grounds. Effect size specification simply means to define the minimum
degree of violation of H0 a researcher would like to detect with a probability not less than (1-").
Cohen’s (1988) definitions of “small”, “medium”, and “large” effects can be helpful in such effect
size specifications (see, e.g., Smith & Bayen, 2005). However, researchers should be aware of the
fact that these conventions may have different meanings for different tests (cf. Erdfelder et al.,
2005).
(3) In compromise power analyses (Erdfelder, 1984; Erdfelder et al., 1996; Müller, Manz,
& Hoyer, 2002), both # and 1-" are computed as functions of the effect size, N, and an error
probability ratio q = " /#. To illustrate, q =1 would mean that the researcher prefers balanced type1 and type-2 error risks (# = "), whereas q = 4 would imply that " = 4 · # (cf. Cohen, 1988).
Compromise power analyses can be useful both before and after data collection. For example, an a
priori power analysis might result in a sample size that exceeds the available resources. In such a
situation, a researcher could specify the maximum affordable sample size and, using a compromise
power analysis, compute # and (1-") associated with, say, q! = " /# = 4. Alternatively, if a study
has already been conducted but not yet been analyzed, a researcher could ask for a reasonable
decision criterion that guarantees perfectly balanced error risks (i.e. # = "), given the size of this
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sample and a critical effect size she is interested in. Of course, compromise power analyses can
easily result in unconventional significance levels larger than # = .05 (in case of small samples or
effect sizes) or less than # = .001 (in case of large samples or effect sizes). However, we believe
that the benefit of balanced type-1 and type-2 error risks often offsets the costs of violating
significance level conventions (cf. Gigerenzer, Kraus, & Vitouch, 2004).
(4) In sensitivity analyses the critical population effect size is computed as a function of #,
1-", and N. Sensitivity analyses may be particularly useful for evaluating published research. They
provide answers to questions like “What is the effect size a study was able to detect with a power
of 1-" = .80, given its sample size and # as specified by the author? In other words, what is the
minimum effect size the test was sufficiently sensitive to?” In addition, sensitivity analyses may be
useful before conducting a study to see whether, given a limited N, the size of the effect that can be
detected is at all realistic (or, for instance, way too large to be expected realistically).
(5) Finally, criterion analyses compute # (and the associated decision criterion) as a
function of 1-", the effect size, and a given sample size. Criterion analyses are alternatives to posthoc power analyses after a study has already been conducted. They may be reasonable whenever
the control of # is less important than the control of ". In case of goodness-of-fit tests for statistical
models, for example, it is most important to minimize the "-risk of wrong decisions in favor of the
model (H0). Researchers could thus use criterion analyses to compute the significance level #
compatible with " = .05 for a small effect size.
Whereas G*Power 2 was limited to the first three types of power analysis, G*Power 3 now
covers all five types. Based on the feedback we received from G*Power 2 users, we believe that
any question related to statistical power that occurs in research practice can be translated into one
of these analysis types.
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3) Program Handling
Using G*Power 3 typically involves the following four steps: (1) Select the statistical test
appropriate for your problem, (2) choose one of the five types of power analysis defined in the
previous section, (3) provide the input parameters required for the analysis, and (4) click on
“calculate” to obtain the results.
In the first step, the statistical test is chosen using the distribution-based or the design-based
approach. G*Power 2 users probably have adapted to the distribution-based approach: One first
selects the family of the test statistic (i.e., t-, F-, z-, !2, or exact test) using the “Test family” menu
in the main window. The “Statistical test” menu adapts accordingly, showing a list of all tests
available for the test family. For the two groups t test, for example, one would first select the t
family of distributions and then “Means: Differences between two independent means (two
groups)” in the “Statistical test” menu (see Figure 1). Alternatively, one might use the designbased approach of test selection. With the “Tests” pull-down menu in the top row it is possible to
select (a) the parameter class the statistical test refers to (i.e., correlations, means, proportions,
regression coefficients, variances) and (b) the design of the study (e.g., number of groups,
independent vs. dependent samples, etc.). For example, researchers would select “Means” !
“Two independent groups” to specify the two-groups t test (see Figure 2). The design-based
approach has the advantage that test options referring to the same parameter class (e.g., means) are
located in close proximity, whereas they may be scattered across different distribution families in
the distribution-based approach.
Please insert Figures 1 and 2 about here.
In the second step the “Type of power analysis” menu in the center of the main window
should be used to choose the appropriate analysis type. In the third step, the power analysis input
parameters are specified in the lower left of the main window. To illustrate, an a priori power
analysis for a two groups t test would require a decision between a one-tailed and a two-tailed test,
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a specification of Cohen’s (1988) effect size measure d under H1, the significance level #, the
required power (1-") of the test, and the preferred group size allocation ratio n2/n1. The final step
consists of clicking “Calculate” to obtain the output in the lower right of the main window.
For instance, input parameters specifying a one-tailed t test, a medium effect size of d = .5,
# = .05, (1-") = .95, and an allocation ratio of n2/n1 = 1 would result in a total sample size of
N=176 (88 observation units in each group; see Figures 1 and 2). The noncentrality parameter $
defining the t distribution under H1, the decision criterion to be used (i.e., the critical value of the t
statistic), the degrees of freedom2 of the t test and the actual power value are also displayed. Note
that the actual power will often be slightly larger than the pre-specified power in a priori power
analyses. The reason is that non-integer sample sizes are always rounded up by G*Power to obtain
integer values consistent with a power level not less than the pre-specified one.
In addition to the numerical output, G*Power 3 displays the central (H0) and the noncentral
(H1) test statistic distributions along with the decision criterion and the associated error
probabilities in the upper part of the main window (see Figure 1)3. This supports understanding the
effects of the input parameters and is likely to be a useful visualization tool in the teaching of, or
the learning about, inferential statistics. The distributions plot may be printed, saved, or copied by
clicking the right mouse button inside the plot area.
The input and output of each power calculation in a G*Power session is automatically
written to a protocol that can be displayed by selecting the “Protocol of power analyses” tab in the
main window. It is possible to clear the protocol, or to print, save, and copy the protocol in the
same way as the distributions plot.
Because Cohen’s (1988) book on power analysis appears to be well known in the social and
behavioral sciences, we made use of his effect size measures whenever possible. Researchers not
familiar with these measures and users preferring to compute Cohen’s measures from more basic
parameters can click on the “Determine” button to the left the effect size input field (see Figures 1
and 2). A drawer will open next to the main window and provide access to an effect size calculator
tailored to the selected test (see Figure 2). For the two-groups t test, for example, users can specify
the means (µ1, µ2) and the common standard deviation (%) in the populations underlying the groups
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to calculate Cohen’s d = |µ1-µ2|/%. Clicking the “Calculate and transfer to main window” button
copies the computed effect size to the appropriate field in the main window.
Please insert Figure 3 about here.
Another useful option is the Power Plot window (see Figure 3) which is opened by clicking the
“X-Y plot for a range of values” in the lower right corner of the main window (see Figures 1 and
2).
By selecting the appropriate parameters for the y- and the x-axis, one parameter (#, power
(1–"), effect size, or sample size) can be plotted as a function of any other parameter. Of the
remaining two parameters, one can be chosen to draw a family of graphs, while the fourth
parameter is kept constant. For instance, power (1–") can be drawn as a function of the sample size
for several different population effects sizes, keeping # at a particular value. The plot may be
printed, saved, or copied by clicking the right mouse button inside the plot area. Selecting the
“table” tab reveals the data underlying the plot; they may be copied to other applications.
The Power Plot window inherits all input parameters of the analysis that is active when the
“X-Y plot for a range of values” button is pressed. Only some of these parameters can be directly
manipulated in the Power Plot window. For instance, switching from a plot of a two-tailed test to
that of a one-tailed test requires choosing the “Tail(s): One” option in the main window, followed
by pressing the “X-Y plot for a range of values” button.
4) Types of Statistical Tests.
G*Power 3 provides power analyses for test statistics following t-, F-, !2- or standard normal
distributions under H0 (either exact or asymptotic) and noncentral distributions of the same test
families under H1. In addition, it includes power analyses for some exact tests. In Tables 2 to 9 we
briefly describe the tests currently covered by G*Power 3. Table 1 lists the symbols and their
meanings as used in Tables 2 to 9.
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Tests for Correlation and Regression
Table 2 summarizes the procedures supported for testing hypotheses on correlation and regression.
One-sample tests are provided for the point-biserial model4, i.e. correlations between a binary
variable and a continuous variable, and for correlations between two normally distributed variables
(Cohen, 1988, Chapter 3). The latter test uses the exact sample correlation coefficient distribution
(Barabesi & Greco, 2002) or, optionally, a large sample approximation based on Fisher’s r-to-z
transformation. The two-sample test for differences between two correlations uses Cohen’s effect
size q (Cohen, 1988, Chapter 4) and is based on Fisher’s r-to-z transformation. Cohen defines q =
.10, q = .30, and q = .50 as “small”, ”medium”, and ”large” effects, respectively.
The two procedures available for the multiple regression model handle the cases of (a) a
test of an overall effect, i.e. the hypothesis that the population value of R2 is different from zero,
and (b) a test of the hypothesis that adding additional predictors increases the value of R2 (Cohen,
1988, Chapter 9). According to Cohen’s criteria effects of size f 2 = 0.02, f 2 = 0.15, and f 2 = 0.35
are considered “small”, ”medium”, and ”large”, respectively.
Tests for Means (univariate case)
Table 3 summarizes the power analysis procedures for tests on means. G*Power 3 supports all
cases of the t test for means described by Cohen (1988, Chapter 2): the test for independent means,
the test of the null hypothesis that the population mean equals some specified value (one sample
case), and the test on the means of two dependent samples (matched pairs). Cohen’s d and dz are
used as effect size indices. Cohen defines d = 0.2, d = 0.5, and d = 0.8 as “small”, ”medium”, and
”large” effects, respectively. Effect size dialogs are available to compute the appropriate effect size
parameter from means and standard deviations. For example, assume we want to compare visual
search times for targets embedded in rare versus frequent local contexts in a within-subject design
(cf. Hoffmann & Sebald, 2005, Exp. 1). It is expected that the mean search time for targets in rare
contexts (say, 600 ms) should decrease by at least 10 ms in frequent contexts (i.e., to 590 ms) as a
consequence of local contextual cuing. If prior evidence suggests population standard deviations
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of, say, % = 25 ms in each of the conditions and a correlation of & = .70 between search times in
both conditions we can use the effect size drawer of G*Power 3 for the matched pairs t test to
calculate the effect size dz = .516 (see Table 3, row 2, for the formula). By selecting a post hoc
power analysis for one-tailed matched pairs t tests, we easily see that for dz = .516, # = .05, and N
= 16 participants the power is only 1-" = .47. Thus, provided that the above assumptions are
appropriate, the nonsignificant statistic t(15) = 1.475 obtained by Hoffmann and Sebald (2005,
Exp. 1, p. 34) might in fact be due to a type 2 error. This interpretation would be consistent with
the fact that Hoffmann and Sebald (2005) observed significant local contextual cuing effects in all
other four experiments they reported.
The procedures provided by G*Power 3 to test effects in between-subjects designs with
more than two groups (i.e., one-way ANOVA designs and general main effects and interactions in
factorial ANOVA designs of any order) are identical to those in G*Power 2 (Erdfelder et al.,
1996). In all these cases the effect size f as defined in Cohen (1988) is used. In a one-way ANOVA
the effect size drawer can be used to compute f from the means and group sizes of k groups and a
standard deviation common to all groups. For tests of effects in factorial designs, the effect size
drawer offers the possibility to compute the effect size f from the variance explained by the tested
effect and the error variance. Cohen defines f = 0.1, f = 0.25, and f = 0.4 as “small”, ”medium”, and
”large” effects, respectively.
New in G*Power 3 are procedures for analyzing main effects and interactions for A ! B
mixed designs, where A is a between-subjects factor (or an enumeration of the groups generated by
cross-classification of several between-subject factors) and B is a within-subjects factor (or an
enumeration of the repeated measures generated by cross-classification of several within-subject
factors). Both the univariate and the multivariate approach to repeated measures (O’Brien &
Kaiser, 1985) are supported. The multivariate approach will be discussed below. The univariate
approach is based on the sphericity assumption. This assumption is correct if (in the population) all
variances of the repeated measurements are equal and all correlations between pairs of repeated
measurements are equal. If all the distributional assumptions are met, then the univariate approach
is the most powerful method (Muller & Barton, 1989; O’Brien & Kaiser, 1985). Unfortunately,
especially the assumption of equal correlations is often violated, which can lead to very misleading
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results. In order to compensate for such adverse effects in tests of within effects or between-within
interactions, the noncentrality parameter and the degrees of freedom of the F-distribution can be
multiplied by a correction factor ' (Geisser & Greenhouse, 1958; Huynh & Feldt, 1970). ' is 1 if
the sphericity assumption is met and approaches 1/(m-1) with increasing degrees of violation of
sphericity, where m denotes the number of repeated measurements.
G*Power provides three separate yet very similar routines to calculate power in the
univariate approach for between effects, within effects, and interactions. If the to-be-detected
effect size f is known, these procedures are very easy to apply. To illustrate, Berti, Münzer,
Schröger, and Pechmann (2006) compared the pitch discrimination ability of 10 musicians and 10
control subjects (between-subjects factor A) for 10 different interference conditions (within-subject
factor B). Assuming that A, B, and A ! B effects of “medium” size (f = .25, cf. Cohen, 1988; Table
3) should be detected given a correlation of & = .50 between repeated measures and a significance
level of # = .05, the power values of the F tests for the A main effect, the B main effect, and the
A ! B interaction are easily computed as .30, .95, and .95, respectively, by inserting f = .25,
# = .05, the total sample size (20), the number of groups (2), the number of repetitions (10) and &
= .50 into the appropriate input fields of the procedures designed for these tests.
If the to-be-detected effect size f is unknown, it is necessary to compute f from more basic
parameters characterizing the expected population scenario under H1. To demonstrate the general
procedure we will show how to do post-hoc power analyses in the scenario illustrated in Figure 4
assuming the variance and correlations structure defined in matrix SR1. We first consider the
power of the within effect: We select the “F tests” family, the “ANOVA: Repeated measures,
within factors” test, and “Post hoc” as the type of power analysis. Both the “Number of groups”
and the “Repetitions” fields are set to 3. The “Total sample size” is set to 90 and the “# error
probability” to 0.05. Referring to matrix SR1, we insert 0.3 in the input field “corr among rep
measures”, and -- since sphericity obviously holds in this case -- we set the “nonsphericity
correction '” to 1. To determine the effect size f, we first calculate ! µ2 , the variance of the within
effect. From the three column means µ • j of matrix M and the grand mean µ •• we get ! µ2 = ((1012.889)2 + (13-12.889)2 + (15.667-12.889)2)/3 = 5.35679. Clicking the “Determine” button next to
the effect size label opens the effect size drawer. We choose “From variances” and set the
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“Variance explained by special effect” to 5.357 and the “Variance within groups” to 92 = 81.
Clicking the “Calculate and transfer to main window” button calculates an effect size f = 0.2572
and transfers f to the effect size field in the main window. Clicking “Calculate” yields the results:
The power is 0.997, the critical F value with df1=2 and df2=174 is 3.048, and the noncentrality
parameter ( is 25.52. The procedure for tests of between-within interactions effects (“ANOVA:
Repeated measures, within-between interaction”) is almost identical to that just described. The
only difference is in how the effect size f is computed: Here we first calculate the variance of the
residual values µ ij ! µ i• ! µ • j + µ •• of matrix M: ! µ2 = ((10-10-15+12.889)2 + … + (12-15.66711.333 + 12.889)2)/9.0 = 1.90123. Using the effect size drawer in the same way as above we get an
effect size f = 0.1532 which results in a power of 0.653. To test between effects we choose
“ANOVA: Repeated measures, between factors” and set all parameters to the same values as
before. Note that in this case we do not need to specify '—no correction is necessary because tests
of between factors do not require the sphericity assumption. To calculate the effect size, we use
“Effect size from means” in the effect size drawer. We select 3 groups, set “SD % within each
group” to 9, and insert for each group the corresponding row mean µ i• of M (15,12.3333,
11.3333) and an equal group size of 30. An effect size f = 0.1719571 is calculated and the resulting
power is 0.488.
Note that G*Power 3 can easily handle pure repeated measures designs without any
between-subject factors (e.g., Frings & Wentura, 2005; Schwarz & Müller, 2006) by choosing the
“ANOVA: Repeated Measures, within factors” procedure and setting the number of groups to 1.
Tests for Means (multivariate case)
G*Power 3 contains several procedures for performing power analyses in multivariate designs (see
Table 3). All these tests belong to the F test family.
The Hotelling T2 tests are extensions of univariate t tests to the multivariate case, where
more than one dependent variable is measured: Instead of two single means two mean vectors are
compared, and instead of a single variance, a variance-covariance matrix is considered (Rencher,
1998). In the one-sample case H0 posits that the vector of population means is identical to a
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specified constant mean vector. The effect size drawer can be used to calculate the effect size
r r
) from the difference µ ! c and the expected variance-covariance matrix under H1. For example,
r r
assume that we have two variables, a difference vector µ ! c = {1.88, 1.88} under H1, variances
! 12 = 56.79, ! 22 = 29.28, and a covariance of 11.98 (Rencher, 1998, p. 106). To perform a post hoc
power analysis, choose “F tests”, then “Hotellings T2: One group mean vector” and set the analysis
type to “Post hoc”. Enter 2 in the “Response variables” field, and then press the “Determine”
button next to effect size label. In the effect size drawer at “Input method: Means and…” choose
“variance-covariance matrix” and press “Specify/Edit input values”. Under the “Means” tab insert
“1.88” in both input fields; under the “Cov Sigma” tab insert 56.79 and 29.28 in the main diagonal
and 11.98 as off-diagonal element in the lower left cell. Pressing the button “Calculate and transfer
to main window” initiates the calculation of the effect size (0.380) and transfers it to the main
window. For this effect size, # = 0.05, and a total sample size of N = 100 the power amounts to
.9282. The procedure in the two-group case is exactly the same, with the following exceptions.
First, in the effect size drawer two mean vectors have to be specified. Second, the group sizes may
differ.
The MANOVA tests in G*Power 3 refer to the multivariate general linear model (O’Brien
& Muller, 1993; O’Brien & Shieh, 1999): Y = XB + " , where Y is N ! p of rank p; X is N ! r of
rank r ; and the r ! p matrix B contains fixed coefficients. The rows of " are taken to be
independent p-variate normal random
vectors with mean 0 and p ! p positive-definite covariance
!
! = " 0 , where C is c ! r with full
matrix " . The multivariate general linear hypothesis is H0: CBA
row rank, and A is p ! a with full column rank (in G*Power 3, " 0 is assumed to be zero). H0 has
H0 refer to the matrices
! df1 = a ! c degrees of freedom. All tests of the hypothesis
!
˙ T W˙X
˙ )"1 CT ] "1 (CBU " # ) = NH*
H = N(CBU " # 0 )T [C( ˙X
0
!
and
!
E = UT "U( N ! rX ) ,
&& is a q ! q “essence model matrix”, W is a q ! q diagonal matrix containing weights
where X
&& T WX
&& ) (see O’Brien & Shieh, 1999, p.14). Let {! * , K , ! * } be the
w j = n j / N , and X T X = N ( X
1
s
s = min(a, c) eigenvalues of E !1 H * and {!1 , K , ! s } the s eigenvalues of E !1 H /( N ! rX ) , i.e.
"i = "i* N /( N ! rX ) .
G*Power 3 (BSC702)
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G*Power 3 offers power analyses for the multivariate model following either the approach
outlined in Muller and Peterson (1984; Muller, LaVange, Landesmann-Ramey & Ramey,1992) or
alternatively the approach of O’Brien and Shieh (1999; Shieh, 2003). Both approaches
approximate the exact distributions of Wilks U (Rao, 1951), the Hotelling-Lawley T1 (Pillai &
Samson, 1959), the Hotelling-Lawley T2 (McKeon, 1974), and Pillai’s V (Pillai & Mijares, 1959)
by F distributions and are asymptotically equivalent. Table 5 outlines details of both
approximations. The type of statistic (U, T1, T2, V) and the approach (Muller-Peterson or O’BrienShieh) can be selected in an Options dialog that can be evoked by clicking the button “Options” at
the bottom of the main window.
The approach of Muller and Peterson (1984) has found widespread use; for instance, it has
been adopted in the SPSS software package. We nevertheless recommend the approach of O’Brien
and Shieh (1999) because it has a number of advantages: (a) Unlike the method of Muller and
Peterson it provides the exact noncentral F distribution whenever the hypothesis involves at most s
= 1 positive eigenvalues, (b) the approximations for s > 1 eigenvalues are almost always more
accurate then the Muller and Peterson method (the Muller and Peterson method systematically
underestimates power), and (c) it provides a simpler form of the noncentrality parameter, i.e. ( =
(* N, where (* is not a function of the total sample size.
G*Power 3 provides procedures to calculate the power for global effects in a “one-way
MANOVA” and for special effects and interactions in factorial MANOVA designs. These
procedures are the direct multivariate analogues of the ANOVA routines described above. Table 5
summarizes information that is needed in addition to the formulae given above to calculate the
effect size f from hypothesized values for the mean matrix M (corresponding to matrix B in the
model), the covariance matrix *, and the contrast matrix C describing the effect under scrutiny.
The effect size drawer can be used to calculate f from known values of the statistic U, T1, T2, or V.
Note, however, that the transformation of T2 to f depends on the sample size. Thus this test statistic
seems not very well suited for a priori analyses. In line with Bredenkamp and Erdfelder (1985) we
recommend V as the multivariate test statistic.
Another group of procedures in G*Power 3 supports the multivariate approach to power
analyses of repeated measures designs. G*Power provides separate but very similar routines for
G*Power 3 (BSC702)
Page 17
the analysis of between effects, within effects and interactions in simple A ! B designs, where A is
a between-subjects factor and B a within-subjects factor. To illustrate the general procedure we
describe in some detail a post hoc analysis of the within-effect for the scenario illustrated in Fig. 4
assuming the variance and correlations structure defined in matrix SR2. We first choose “F tests”,
then “MANOVA: Repeated measures, within factors”. In the “Type of power analysis” menu we
choose “Post hoc”. We click the “Options” button to open a dialog in which we deselect the “Use
mean correlation in effect size calculation” option. We choose the “Pillai V” statistic and the
“O’Brien and Shieh” algorithm. Back to the main window we set both “Number of groups” and
“Repetitions” to 3, the “Total sample size” to 90, and the “# error probability” to 0.05. To compute
the effect size f(V) for the Pillai statistic we open the effect size drawer by clicking on the
“Determine” button next to the effect size label. In the effect size drawer select, as procedure,
“Effect size from mean and variance-covariance matrix” and, as input method, “SD and correlation
matrix”. Clicking on “Specify/Edit matrices” opens another window in which we specify the
hypothesized parameters. In the “means” tab we insert our means matrix M, in the “Cov Sigma”
tab we choose “SD and Correlation” and insert the values of SR2. Because this matrix is always
symmetric, it suffices to specify the lower diagonal values. After closing the dialog and clicking on
“Calculate and transfer to main window” we get a value 0.1791 for Pillai’s V and the effect size
f(V) = 0.4672. Clicking on “Calculate” shows that the power is 0.980. The analysis of between
effects and interaction effects is done in an analogous way.
Tests for Proportions
The support for tests on proportions has been greatly enhanced in G*Power 3. Table 6 summarizes
the tests that are currently implemented. In particular, all tests on proportions considered by Cohen
(1988) are now available: (a) the sign test (Cohen, 1988, Chapter 5), (b) the z-test for the
difference between two proportions (Cohen, 1988, Chapter 6), and (c) the !2-test for goodness-offit and contingency tables (Cohen, 1988, Chapter 7).
The sign test is implemented as a special case (c = 0.5) of the more general binomial test
(also available in G*Power 3) that a single proportion has a specified value c. In both procedures
G*Power 3 (BSC702)
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Cohen’s effect size g is used and exact power values based on the binomial distribution are
calculated. Note, however, that due to the discrete nature of the binomial distribution the nominal
value of # usually cannot be realized. Since the tables in Chapter 5 of Cohen’s book use the #
value closest to the nominal value, even if it is higher than the nominal value, the tabulated power
values are sometimes larger than those calculated by G*Power 3. G*Power 3 always requires the
actual # not to be larger than the nominal value.
Numerous procedures have been proposed to test the null hypothesis that two independent
proportions are identical (D’Agostino, Chase & Belanger, 1988; Cohen, 1988; Suissa & Shuster,
1985; Upton, 1982), and G*Power 3 implements several of them. The simplest procedure is a z test
with optional arcsin transformation and optional continuity correction. Besides these two
computational options it can also be chosen whether Cohen’s effect size measure h or,
alternatively, two proportions are used to specify the alternate hypothesis. Using the options (“Use
continuity correction” off, “Use arcsin transform” on) the procedure calculates power values close
to those tabulated in Cohen (1988, Chapter 6). The options (“Use continuity correction” off, “Use
arcsin transform” off) compute the uncorrected !2-approximation (Fleiss, 1981) whereas (“Use
continuity correction” on, “Use arcsin transform” off) computes the corrected !2-approximation
(Fleiss, 1981).
A second variant is Fisher’s exact conditional test (Haseman, 1978). Normally, G*Power 3
calculates the exact unconditional power. However, despite the highly optimized algorithm used in
G*Power 3, long computation times may result for large sample sizes (say N > 1000). Therefore a
limiting N can be specified in the Options dialog that determines at which sample size G*Power 3
switches to a large sample approximation.
A third variant calculates the exact unconditional power for approximate test statistics T
(Table 7 summarizes the supported statistics). The logic underlying this procedure is to enumerate
all possible outcomes for the 2 x 2 binomial table, given fixed sample sizes n1, n2 in the two
groups. This is done by choosing as success frequency x1 and x2 in each group any combination of
the values 0 < x1 " n1 and 0 < x2 " n2. Given the success probabilities +1, +2 in each group, the
probability of observing a table X with success frequencies x1, x2 is:
#n &
#n &
P( X | "1 ," 2 ) =% 1 ("1x1 (1) "1 ) n1 )x1 % 2 (" 2x 2 (1) " 2 ) n2 )x 2
$ x1 '
$ x2 '
G*Power 3 (BSC702)
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To calculate power (1-") and the actual type-I error #,, the test statistic T is computed for
each table and compared with the critical value T#. If A denotes the set of all tables X rejected by
this criterion, i.e. those with T > T#, then the power and the # level are given by:
1" # = & X % A P( X | $1 ,$ 2 ) and " * = % X $ A P( X | # 2 ,# 2 ) , where +2 denotes the success probability in
both groups as assumed in the null hypothesis. Please note that the actual # level can be larger that
!
the nominal level! The
! preferred input method (proportions, difference, risk ratio, odds ratio; see
Table 6) and the test statistic to use (see Table 7) can be changed in the Options dialog. Note that
the test statistic actually used to analyze the data should be chosen. For large sample sizes the
exact computation may take too much time. Therefore, a limiting N can be specified in the Options
dialog that determines at which sample size G*Power switches to large sample approximations.
G*Power 3 also provides a group of procedures to test the hypothesis that the
difference/risk ratio/odds ratio of a proportion with respect to a specified reference proportion + is
different under H1 than a difference/risk ratio/odds ratio to the same reference proportion assumed
in H0. These procedures are available in the “exact” test family as “Proportions: Inequality (offset),
two independent groups (unconditional)”. The enumeration procedure described above for the tests
on differences between proportions without offset is also used in this case. In the tests without
offset, the different input parameters (differences, risk ratio, etc.) are equivalent ways of specifying
two proportions. The specific choice has no influence on the results. In the case of tests with offset,
however, each input method has a different set of available test statistics. The preferred input
method (proportions, difference, risk ratio, odds ratio; see Table 6) and the test statistic to use (see
Table 8) can be changed in the Options dialog. As in the other exact procedures, the computation
may be time consuming and a limiting N can be specified in the Options dialog that determines at
which sample size G*Power switches to large sample approximations.
Also new in G*Power 3 is an exact procedure to calculate the power for the McNemar test.
The null hypothesis of this test states that the proportions of successes are identical in two
dependent samples. Figure 5 shows the structure of the underlying design: A binary response is
sampled from the same subject or a matched pair in a standard and in a treatment condition. The
G*Power 3 (BSC702)
Page 20
null hypothesis +s = +t is formally equivalent to the hypothesis for the odd ratio: OR =! 12 / ! 21 = 1 .
To fully specify H1 we not only need to specify the odds ratio, but also the proportion +D of
discordant pairs, that is, the expected proportion of responses that differ in the standard and the
treatment condition. The exact procedure used in G*Power 3 calculates the unconditional power
for the exact conditional test, which calculates the power conditional on the number nD of
discordant pairs. Let p(nD = i) be the probability that the number of discordant pairs is i. Then the
unconditional power is the sum over all i - {0, …. N} of the conditional power for nD = i weighted
with p(nD = i). This procedure is very efficient, but for very large sample sizes the exact
computation nevertheless may take too much time. Again, a limiting N can be specified in the
Options dialog that determines at which sample size G*Power switches to a large sample
approximation. The large sample approximation calculates the power based on an ordinary one
sample binomial test with Bin(N+D, 0.5) as the distribution under H0 and Bin(N+D, OR/(1+OR)) as
the H1 distribution.
Tests for Variances
Table 9 summarizes important properties of the two procedures for testing hypotheses on variances
that are currently supported by G*Power 3. In the one-group case the null hypothesis is tested that
the population variance %2 has a specified value c. The variance ratio %2/c is used as the effect size.
The “central and noncentral” distributions, corresponding to H0 and H1, respectively, are central !2
distributions with N–1 degrees of freedom (because H0 and H1 are based on the same mean). To
compare the variance distributions under both hypotheses, the H1 distribution is scaled with the
value r postulated for the ratio %2/c in the alternate hypothesis, i.e. the noncentral distribution is
r!2N-1 (Ostle & Malone, 1988). In the two-groups case H0 states that the variances in two
populations are identical (%2/%1 = 1). Analogous to the one sample case, two central F distributions
are compared, the H1 distribution being scaled by the value of the variance ratio %2/%1 postulated in
H1.
Generic tests
G*Power 3 (BSC702)
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Besides the specific routines described in Tables 2 to 9 that cover a considerable part of the tests
commonly used, G*Power 3 provides “generic” power analysis routines that may be used for any
test based on the t, F, !2, z, and binomial distribution. In generic routines the parameters of the
central and noncentral distributions are specified directly.
To demonstrate the uses and limitations of these generic routines we will show how to do a
two-tailed power analysis for the one-sample t test by using the generic routine. You may compare
the results with those of the specific routine available in G*Power for that test. First, we select the
“t tests” family and then “Generic t test” (the generic test option is always located at the end of the
list of tests). Next, we select “Post hoc” as the type of power analysis. We choose a two-tailed test
and 0.05 as “# error probability”. We now need to specify the noncentrality parameter $ and the
degrees of freedom for our test. We look up the definitions for the one-sample test in Table 3 and
find " = d N and df = N ! 1 . Assuming a “medium” effect of d = 0.5 and N = 25 we arrive at $ =
0.5·5 = 2.5, and df = 24. Inserting these values and clicking “Calculate” we obtain a power of (1-")
!
= 0.6697. The critical value t = 2.0639 corresponds to the specified #. In this post hoc power
analysis, the generic routine is almost as simple as the specific routine. The main disadvantage of
the generic routines is, however, that the dependence of the noncentrality parameter on the sample
size is implicit. As a consequence, we cannot perform a priori analyses automatically. Rather, we
need to iterate N by hand until we find an appropriate power value.
5) Statistical Methods and Numerical Algorithms
The subroutines used to compute the distribution functions (and the inverse) of the noncentral t, F,
!2, the z and the binomial distribution are based on the C-version of the DCDFLIB (available from
http://www.netlib.org/random/) which was slightly modified for our purposes. G*Power 3 no
longer provides approximate power analyses that were available in the speed mode of G*Power 2.
Two arguments guided us in supporting exact power calculations only. First, four-digit precision of
power calculations may be mandatory in many applications. For example, both compromise power
analyses for very large samples and error probability adjustments in case of multiple tests of
significance may result in very small values of # or " (Westermann & Hager, 1986). Second, as a
G*Power 3 (BSC702)
Page 22
consequence of improved computer technology, exact calculations have become so fast that the
speed gain associated with approximate power calculations is not even noticeable. Thus, from a
computational standpoint, there is little advantage to using approximate rather than exact methods
(cf. Bradley, Russel, & Reeve, 1998).
6) Program Availability and Internet Support
To summarize, G*Power 3 is a major extension of, and improvement over, G*Power 2 in that it
offers easy-to-apply power analyses for a much larger variety of common statistical tests. Program
handling is more flexible, easier to understand, and more intuitive compared to G*Power 2,
reducing the risk of erroneous applications. The added graphical features should be useful for both
research and teaching purposes. Thus, G*Power 3 is likely to become a useful tool for empirical
researchers and students of applied statistics.
Like its predecessor, G*Power 3 is a noncommercial program that can be downloaded free
of charge. Copies of the Mac and the Windows versions are available at http://www.psycho.uniduesseldorf.de/abteilungen/aap/gpower3/ only. Users interested in distributing the program in
another way must ask for permission from the authors. Commercial distribution is strictly
forbidden.
The G*Power 3 webpage will offer an expanding web-based tutorial describing how to use
the program, along with examples. Users who let us know their e-mail addresses will be informed
of updates. Although considerable effort has been put into program development and evaluation,
there is no warranty whatsoever. Users are kindly asked to report possible bugs and difficulties in
program handling to [email protected].
G*Power 3 (BSC702)
Page 23
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G*Power 3 (BSC702)
Page 29
Footnotes
1
The “observed power” is reported in many frequently used computer programs (e.g., the
MANOVA procedure of SPSS).
2
We recommend to check the df’s reported by G*Power, for example, by comparing them to the
df’s reported by the program used to analyze the sample data. If the df’s do not match, the input
provided to G*Power is incorrect and the power calculations do not apply.
3
Plots of the central and non-central distribution are only shown for tests based on the t-, F-, z-, !2,
or binomial distribution. No plots are shown for tests that involve an enumeration procedure (e.g.
the McNemar test).
4
We would like to thank Dave Kenny for making us aware of the fact the t test (correlation) power
analyses of G*Power 2 are correct only in the point-biserial case (i.e., correlations between a
binary variable and continuous variable, the latter being normally distributed for each value of the
binary variable). For correlations between two continuous variables following a bivariate normal
distribution, the t test (correlation) procedure of G*Power 2 overestimates power. For this reason,
G*Power 3 offers separate power analyses for point-biserial correlations (in the t family of
distributions) and correlations between two normally distributed variables (in the exact distribution
family). However, power values will usually differ only slightly between procedures. To illustrate,
assume we are interested in the power of a two-tailed test of H0: & = .00 for continuously
distributed measures derived from two Implicit Association Tests (IATs) differing in content.
Assume further that, due to method-specific variance in both versions of the IAT, the true Pearson
correlation is actually & = .30 (effect size). Given # = .05 and N = 57 (see Back, Schmukle, &
Egloff, 2005, Study 3, p. 173), an exact post-hoc power analysis for “Correlations: Differences
from constant (one sample case)” reveals the correct power value of (1-") = .63. Choosing the
G*Power 3 (BSC702)
Page 30
incorrect “Correlation: point biserial model” procedure from the t test family would result in (1-")
= .65.
G*Power 3 (BSC702)
Page 31
Authors Note
Franz Faul, Institut für Psychologie, Christian-Albrechts-Universität, Kiel, Germany; Edgar
Erdfelder, Lehrstuhl für Psychologie III, Universität Mannheim, Mannheim, Germany; AlbertGeorg Lang and Axel Buchner, Institut für Experimentelle Psychologie, Heinrich-HeineUniversität, Düsseldorf, Germany.
Manuscript preparation has been supported by grants from the Deutsche
Forschungsgemeinschaft (SFB 504, Project A12) and the state of Baden-Württemberg, Germany
(Landesforschungsprogramm “Evidenzbasierte Stressprävention”).
Correspondence concerning this article should be addressed to Franz Faul, Institut für
Psychologie, Christian-Albrechts-Universität, Olshausenstr. 40, D-24098 Kiel, Germany (email:
[email protected]), or Edgar Erdfelder, Lehrstuhl für Psychologie III, Universität
Mannheim, Schloss, D-68131 Mannheim, Germany (email: [email protected]).
G*Power 3 (BSC702)
32
Page
Figure Captions
Figure 1:
The distribution-based approach of test specification in G*Power 3.0
Figure 2:
The design-based approach of test specification in G*Power 3.0 and the effect size drawer
Figure 3:
The Power Plot window of G*Power 3.0
Figure 4:
Sample 3 x 3 Repeated Measures designs: Three groups are repeatedly measured at three different
times. The shaded portion of the table is the postulated matrix M of population means µij. The last
column of the table contains the sample size in each group. The symmetric matrices SRi specify
two different covariance structures between measurements taken at different times: The main
diagonal contains the standard deviations of the measurements at each time, the off-diagonal
elements contain the correlations between pairs of measurements taken at different times.
Figure 5:
Matched binary response design (McNemar test).
G*Power 3 (BSC702)
Page 33
Figure 1
G*Power 3 (BSC702)
Page 34
Figure 2
G*Power 3 (BSC702)
Page 35
Figure 3
G*Power 3 (BSC702)
Page 36
Figure 4
Group 1
Group 2
Group 3
µ• j
Time 1
Time 2
Time 3
µ i•
ni
10
10
10
10
15
12
12
13
20
15
12
15.667
15
12.333
11.333
30
30
30
µ•• = 12.889
& 10 0.3 0.1#
$
!
SR 2 = $ 0.3 9 0.3 !
$ 0.1 0.3 8 !
%
"
& 9 0.3 0.3 #
$
!
SR1 = $ 0.3 9 0.3 !
$ 0.3 0.3 9 !
%
"
G*Power 3 (BSC702)
Page 37
Figure 5
Treatment
Yes
No
!
!
!
Standard
Yes
No
"11
"12
" 21
" 22
"s
1" # s
!
!
!
!
!
Proportion of discordant pairs: " D = "12 + " 21
"t
1" # t
1
Hypothesis: ! s = ! t or equivalently "12 = " 21
!
!
Tabl% 1( )ymb,ls and th%ir m%anings as us%d in th% tabl%s in this articl%7
)ymb,ls
8%aning
! ; : !i 9
<,pulati,n m%an :in gr,up i9
! ; : !i 9
!
V%ct,r ,f p,pulati,n m%ans :in gr,up i9
!x" y
<,pulati,n m%an ,f th% diff%r%nc%
N
T,tal sampl% siz%
ni
)ampl% siz% in gr,up i
#
)tandard d%viati,n in th% p,pulati,n
#!
)tandard d%viati,n ,f th% %ff%ct
# x" y
)tandard d%viati,n ,f th% diff%r%nc%
$
N,nc%ntrality param%t%r ,f th% n,nc%ntral F and !2 distributi,n
%
N,nc%ntrality param%t%r ,f th% n,nc%ntral t distributi,n
df
D%gr%%s ,f fr%%d,m
df1 ; df 2
Num%rat,r E d%n,minat,r d%gr%%s ,f fr%%d,m
& ; : &i 9
<,pulati,n c,rr%lati,n :in gr,up i9
RY2' A ; RY2' A; B
)quar%d multipl% c,rr%lati,n c,%ffici%nts; c,rr%sp,nding t, th% pr,p,rti,n ,f Y
varianc% that can b% acc,unt%d f,r by multipl% r%gr%ssi,n ,n th% s%t ,f pr%dict,r
variabl%s A and A ( B ; r%sp%ctiv%ly
!
<,pulati,n varianc%-c,varianc% matrix
M
8atrix ,f r%gr%ssi,n param%t%rs :p,pulati,n m%ans9
C
I,ntrast matrix :c,ntrasts b%tw%%n r,ws ,f M 9
A
I,ntrast matrix :c,ntrasts b%tw%%n c,lumns ,f M 9
) ; :) i 9
<r,p,rti,n :pr,bability9 ,f succ%ss :in gr,up i9
!
Tabl% 2( T%sts f,r c,rr%lati,n and r%gr%ssi,n
I,rr%lati,n and R%gr%ssi,n
T%st
T%st
family
Null
Hyp,th%sis
Eff%ct )iz%
Diff%r%nc% fr,m
z%r,( <,int
bis%rial m,d%l
t t%sts
& *O
&
Diff%r%nc% fr,m
c,nstant
:bivariat% n,rmal9
%xact
& *c
&
Pn%quality ,f tw,
c,rr%lati,n
c,%ffici%nts
z t%sts
&1 * & 2
q * z1 " z 2
Nth%r
<aram%t%rs
N,nc%ntrality <aram%t%r
% *
&2
' N
1" & 2
df * N " 2
I,nstant
c,rr%lati,n c
zi *
8ultipl%
R%gr%ssi,n(
d%viati,n ,f R2
fr,m z%r,
F t%sts
8ultipl%
R%gr%ssi,n(
incr%as% ,f R2
F t%sts
RY2'A * O
RY2' A; B * RY2' A
1 1 + &i
ln
2 1 " &i
f2 *
f
2
*
m1 *
q
s
s*
n1 + n 2 " R
:n1 " Q9:n2 " Q9
RY2' A
1 " RY2' A
Numb%r ,f
pr%dict,rs p
:SA9
$ * f 2N
RY2' A; B " RY2' A
T,tal numb%r ,f
pr%dict,rs p
:SA U SB9
$ * f 2N
1 " RY2' A; B
Numb%r ,f
t%st%d pr%dict,rs
q :SB9
df1 * p
df 2 * N " p " 1
df1 * q
df 2 * N " p " 1
Tabl% Q( T%sts f,r univariat% m%ans
8%ans :univariat%9
T%st
T%st
Family
Null
Hyp,th%sis
Diff%r%nc%
fr,m c,nstant
:,n% sampl%
cas%9
t t%sts
! *c
Pn%quality ,f
tw, d%p%nd%nt
m%ans
:match%d pairs9
t t%sts
! x" y * O
Eff%ct )iz%
d*
Nth%r
<aram%t%rs
! "c
#
dz *
N,nc%ntrality <aram%t%r
and D%gr%%s ,f Fr%%d,m
% *d N
df * N " 1
X ! x" y X
# x "7 y
% * dz N
;
df * N " 1
# x" y *
# x2 + # y2 " 2 &# x# y
Pn%quality ,f
tw,
ind%p%nd%nt
m%ans
t t%sts
!1 * ! 2
ANNVA; fix%d
%ff%cts; ,n%way( Pn%quality
,f multipl%
m%ans
F t%sts
!i " ! * O;
i * 1;"; k
!1 " ! 2
#
f *
#!
;
#
n1n2
n1 + n2
df * N " 2
% *d
k
# !2 *
ANNVA; fix%d
%ff%cts; multifact,r d%signs
and plann%d
c,mparis,ns
F t%sts
ANNVA(
r%p%at%d
m%asur%m%nts;
b%tw%%n %ff%cts
F t%sts
ANNVA(
r%p%at%d
m%asur%m%nts
within %ff%cts
d*
!i " ! * O;
i * 1;"; k
f *
, n :! " ! 9
j
Numb%r ,f
gr,ups k
$ * f 2N
T,tal numb%r ,f
c%lls in th%
d%sign k
$ * f 2N
2
i
i *1
#!
#
N
df1 * q
df 2 * : N " k 9
D%gr%%s ,f
fr%%d,m ,f th%
t%st%d %ff%ct q7
!i " ! * O;
i * 1;"; k
F t%sts
!i " ! * O;
i * 1;"; m
f *
f *
#!
#
#!
#
Y%v%ls ,f th%
b%tw%%n fact,r
k
Y%v%ls ,f th%
r%p%at%d
m%asur% fact,r
m
<,pulati,n
c,rr%lati,n
am,ng r%p%at%d
m%asur%m%nts
&
ANNVA(
r%p%at%d
m%asur%m%nts
b%tw%%n-within
int%racti,ns
df1 * k " 1
df 2 * N " k
F t%sts
!ij " !i " #
! j + ! * O;
i * 1;"; k
j * 1;"; m
f *
#!
#
F,r within and
within-b%tw%%n
int%racti,ns(
N,nsph%ricity
c,rr%cti,n !
1
. - .1
m "1
$ * f 2uNu*
m
1 + :m " 19 &
df1 * :k " 19
df 2 * : N " k 9
$ * f 2uN
m
1" &
df1 * :m " 19-
u*
df 2 * : N " k 9:m " 19-
$ * f 2uNm
1" &
df1 * :k " 19:m " 19-
u*
df 2 * : N " k 9:m " 19-
Tabl% Z( T%sts f,r multivariat% m%ans
8%ans :multivariat%9
T%st
H,t%lling T2
diff%r%nc% fr,m
c,nstant m%an
v%ct,r
H,t%lling T2
diff%r%nc%
b%tw%%n tw,
m%an v%ct,rs
8ANNVA
gl,bal %ff%cts
T%st
Family
F t%sts
F t%sts
Null
Hyp,th%sis
! !
! *c
!
!
!1 * ! 2
!
!
/ * v T ! "1v
! !
v * ! "c
Nth%r
<aram%t%rs
Numb%r ,f
r%sp,ns%
variabl%s k
!
!
/ * v T ! "1v
! !
v * !1 " ! 2
Numb%r ,f
r%sp,ns%
variabl%s k
N,nc%ntrality <aram%t%r
and D%gr%%s ,f Fr%%d,m
$ * /2 N
df1 * k
df 2 * N " k
$ * /2
n1n2
n2 + n2
df1 * k
df 2 * N " k " 1
F t%sts
CM * 0
8%ans matrix
M
8ANNVA
sp%cial %ff%cts
Eff%ct )iz%
F t%sts
I,ntrast
matrix C
Eff%ct siz%
f mult
d%p%nds ,n th% t%st
statistics(
" [ilks U
" H,t%lling-Yawl%y T1
" H,t%lling-Yawl%y T2
" <illai V
and alg,rithms(
8ANNVA(
r%p%at%d
m%asur%m%nts;
b%tw%%n %ff%cts
F t%sts
8ANNVA(
r%p%at%d
m%asur%m%nts
within %ff%cts
F t%sts
8ANNVA(
r%p%at%d
m%asur%m%nts
b%tw%%n-within
int%racti,ns
F t%sts
CMA * 0
8%ans matrix
M
B%tw%%n
c,ntrast
matrix C
[ithin
c,ntrast
matrix A
" 8ull%r and <%t%rs,n
:1^_Z9
" N`Bri%n and )hi%h
:1^^^9
Numb%r ,f
gr,ups g
Numb%r ,f
r%sp,ns%
variabl%s k
Numb%r ,f
gr,ups g
Numb%r ,f
pr%dict,rs p
Numb%r ,f
r%sp,ns%
variabl%s k
Y%v%ls ,f th%
b%tw%%n fact,r
k
Y%v%ls ,f th%
r%p%at%d
m%asur% fact,r
m
N,nc%ntrality param%t%r
and d%gr%%s ,f fr%%d,m
d%p%nd ,n th% t%st
statistic and alg,rithm
us%d :s%% %ff%ct siz%
c,lumn and Tabl% R97
Tabl% a( Appr,ximating univariat% statistics f,r multivariat% hyp,th%s%s
Appr,ximating univariat% statistics
)tatistic
F,rmula
[ilks U
8<
U*
Num%rat,r D%gr%% ,f Fr%%d,m df2
s
0 :1 + 1 9
"1
k
k *1
df 2 * g : N " g1 9 " g 2 c
g1 * r + :a " c + 19 E 2
g 2 * :ca " 29 E 2
[ilks U
N)
<illai V
8<
U*
s
0 :1 + 1 9
b "1
k
k *1
V*
s
01
k
E:1 + 1k 9
1
5
2
g * 4 :ca9 2 " Z
2 c2 + a2 " a
3
Eff%ct )iz% and N,nc%ntrality
<aram%t%r
f :U 9 2 *
1 " U 1E g
U 1E g
$ * f :U 9 2 df 2
ca . Q
ca 6 Z
df 2 * s : N " r " a + s 9
f :U 9 2 *
1 " U 1E g
U 1E g
$ * Ng f :U 9 2
f :V 9 2 *
k *1
V
:s " V 9
$ * f :V 9 2 df 2
<illai V
N)
V*
s
01
b
k
f :V 9 2 *
E:1 + 1kb 9
k *1
V
:s " V 9
$ * Nsf :V 9 2
H,t%llingYawl%y T1
8<
T*
H,t%llingYawl%y T1
N)
T*
H,t%llingYawl%y T2
8<
T*
H,t%llingYawl%y T2
N)
T*
df 2 * s : N " r " a " 19 + 2
s
01
k
$ * f :T 9 2 df 2
k *1
s
01
f :T 9 2 * T E s
b
k
$ * Nsf :T 9 2
k *1
s
0 1k
k *1
s
01kb
k *1
f :T 9 2 * T E s
df 2 * Z + :ca + 29 g
g*
: N " r 92 " : N " r 9 g Z + gQ
: N " r 9 g 2 " g1
g1 * c + 2a + a 2 " 1
g2 * c + a + 1
g Q * a:a + Q9
g Z * 2a + Q
h * :df 2 " 29 E: N " r " a " 19
f :T 9 2 * T E h
$ * f :T 9 2 df 2
f :T 9 2 * T E h
$ * Nhf :T 9 2
Tabl% R( T%sts f,r pr,p,rti,ns
<r,p,rti,ns
T%st
T%st
Family
Hyp,th%sis
I,nting%ncy
tabl%s d
e,,dn%ss ,f Fit
!2 t%sts
) 1i * ) Oi
i * 1;"; k
,
k
i *1
Eff%ct )iz%
k
,
w*
Nth%r <aram%t%rs
N,nc%ntrality
<aram%t%r
$ * w2 N
:) 1i " ) Oi 9 2
i *1
) Oi
) Oi * 1
Diff%r%nc% fr,m
c,nstant :,n%
sampl% cas%9
%xact
) *c
g *) "c
I,nstant pr,p,rti,n c
Pn%quality ,f tw,
d%p%nd%nt
pr,p,rti,ns
:8cN%mar9
%xact
) 12 E) 21 * 1
Ndds rati, * ) 12 E ) 21
<r,p,rti,n ,f disc,rdant
pairs * ) 12 + ) 21
)ign t%st
%xact
) * 1E 2
g * ) " 1E 2
Pn%quality ,f tw,
ind%p%nd%nt
pr,p,rti,ns
z t%sts
)1 * ) 2
:A9 Alt%rnat% pr,p7 ) 2 ;
:A9 Null pr,p7 ) 1
:B9 h * 11 " 12
1i * 2 arcsin ) i
Pn%quality ,f tw,
ind%p%nd%nt
pr,p,rti,ns
:Fish%r`s %xact
t%st9
%xact
)1 * ) 2
alt%rnat% pr,p7 ) 1
Null pr,p7 ) 2
Pn%quality ,f
tw, ind%p%nd%nt
pr,p,rti,ns
:unc,nditi,nal9
%xact
)1 * ) 2
:A9 Alt%rnat% pr,p7( ) 1
:B9 Diff%r%nc%( ) 2 " ) 1
:I9 Risk rati,( ) 2 E ) 1
:D9 Ndds rati,(
) 1 E:1 " ) 1 9
) 2 E:1 " ) 2 9
Null pr,p7 ) 2
Pn%quality with
,ffs%t ,f tw,
ind%p%nd%nt
pr,p,rti,ns
:unc,nditi,nal9
%xact
)1 * ) 2 + c
:A9 Alt%rnat% pr,p( ) 1X H 1
:A9 <r,p7( ) 1X H
:B9 Diff%r%nc%( ) 2 " ) 1X H
:B9 Diff%r%nc%( ) 2 " ) 1X H
:I9 Risk rati,( ) 2 E ) 1X H
:D9 Ndds rati,(
) 1X H E:1 " ) 1X H 9
1
) 2 E:1 " ) 2 9
1
1
1
O
:I9 Risk rati,( ) 2 E ) 1X H
:D9 Ndds rati,(
) 1X H E:1 " ) 1X H 9
O
) 2 E:1 " ) 2 9
Null pr,p7 ) 2
O
O
O
Tabl% f( T%st statistics us%d in t%sts ,f th% diff%r%nc% b%tw%%n tw, ind%p%nd%nt pr,p,rti,ns7 Th%
z-t%sts in th% tabl% ar% m,r% c,mm,nly kn,wn as 72 t%sts :th% %quival%nt z t%st is us%d t, pr,vid%
tw,-sid%d t%sts97
Nr
Nam%
1
z-t%st p,,l%d varianc%
2
z-t%st p,,l%d varianc%
with c,ntinuity
c,rr%cti,n
Q
z-t%st unp,,l%d varianc%
Z
z-t%st unp,,l%d varianc%
with c,ntinuity
c,rr%cti,n
a
8ant%l-Ha%nsz%l t%st
)tatistic
z*
z*
z*
z*
z*
=1 1:
n )g + n )g
)g1 " )g 2
c #g * )g :1 " )g 9;; + 88 c )g * 1 1 2 2
#g
n1 + n2
< n1 n2 9
k=1 1 :
+ 8
2 < n1 n2 89
5" 1 l,w%r tail
c #g s%% :19c k * 4
g
#
3+ 1 upp%r tail
)g1 " )g 2 + ;;
)g1 " )g 2
)g1 :1 " )g1 9 )g 2 :1 " )g 2 9
+
c #g *
#g
n1
n2
k=1 1 :
+ 8
2 < n1 n2 89
5" 1 l,w%r tail
c #g s%% :Q9c k * 4
g
#
3+ 1 upp%r tail
)g1 " )g 2 + ;;
x1 " E : x1 9
V : x1 9
c E : x1 9 *
n1 : x1 + x2 9
n n : x + x 9: N " x1 " x2 9
c V : x1 9 * 1 2 1 2 2
N
N : N " 19
R
Yik%lih,,d rati,
:Upt,n; 1^_29
= t : x1 9 + t : x2 9 + t :1 " x1 9 + t :1 " x2 9 + t : N 9 :
8 c t : x9 (* x ln: x9
lr * 2;;
8
"
"
"
+
"
"
"
t
n
t
n
t
x
x
t
N
x
x
:
9
:
9
:
9
:
9
1
2
1
2
1
2
<
9
f
t t%st with df = N-2
:D`Ag,stin, %t al; 1^__9
t N " 2 * : x1 :1 " x2 9 " x2 :1 " x1 99
N "2
N jn2 x1 :1 " x1 9 + n1 x2 :1 " x2 9i
N,t%--- xi = succ%ss fr%qu%ncy in gr,up ic ni = sampl% siz% in gr,up ic N * n1 + n2 = t,tal sampl% siz%c )gi * xi E ni
Tabl% _( T%st statistics us%d in t%sts ,f th% diff%r%nc% with ,ffs%t b%tw%%n tw, ind%p%nd%nt
pr,p,rti,ns7
Nr
Nam%
)tatistic
1
z-t%st p,,l%d
varianc%
z*
)g1 " )g 2 " %
n )g + n )g
c #g * )g :1 " )g 9>1 E n1 + 1 E n2 ? c )g * 1 1 2 2
#g
n1 + n2
2
z-t%st p,,l%d
varianc% with
c,nt7c,rr7
z*
5" 1 l,w%r tail
)g1 " )g 2 " % + k E 2>1 E n1 + 1 E n2 ?
c #g s%% :19c k * 4
#g
3+ 1 upp%r tail
Q
z-t%st unp,,l%d
varianc%
z*
)g1 " )g 2 " %
c #g * )g1 :1 " )g1 9 E n1 + )g 2 :1 " )g 2 9 E n2
#g
Z
z-t%st unp,,l%d
varianc% with
c,nt7c,rr7
z*
5" 1 l,w%r tail
)g1 " )g 2 " % + k E 2>1 E n1 + 1 E n2 ?
c #g s%% :Q9c k * 4
#g
3+ 1 upp%r tail
a
t t%st with df = N-2 t N " 2 * :: x1 + % n1 9:1 " x2 9 " x2 :1 " x1 " % n1 99 K c
:D`Ag,stin, %t al;
K * : N " 29 ElN jn2 x1 :1 " x1 9 + n1 x2 :1 " x2 9ik
1^__9
R
Yik%lih,,d sc,r%
rati, :diff%r%nc%9
:8i%ttin%n d
Nurmin%n; 1^_a9
Farringt,n d
8anning :1^^O9
eart d Nam
:1^^O9
)g1 " )g 2 " %
c #g * >)m1 :1 " )m1 9 E n1 + )m2 :1 " )m2 9 E n2 ?K
#g
8i%ttin%n d Nurmin%n( K = NE:N-19; Farringt,n d 8anning( K =1
z*
)m1 * 2u c,s:w9 " b E:Qa9 c )m2 * )m1 " % c
@ * n2 E n1 c a * 1 + @ c b * ":1 + @ + )g1 + @)g 2 + % :@ + 299 c
c * % 2 + % :2)g1 + @ + 19 + )g1 + @)g 2 c d * ")g1% :1 + % 9c
v * bQ E:Qa 9Q " bc E:Ra 2 9 + d E: 2a 9 c w * :Q71Z1a^ + c,s "1 :v E u Q 99 E Q c
u * sgn:v9 b 2 E:Qa9 2 " c E:Qa9
)k%wn%ss c,rr%ct%d z n :eart d Nam9c z acc,rding t, Farringt,n d 8anning(
"1
z n * 1 + ZA :A + z 9 " 1 E 2A c V * >)m :1 " )m 9 E n + )m :1 " )m 9 E n ? c
>
A *V
f
Yik%lih,,d sc,r%
rati, :risk rati,9
:8i%ttin%n d
Nurmin%n; 1^_a9
Farringt,n d
8anning :1^^O9
eart d Nam
:1^__9
_
Yik%lih,,d sc,r%
rati, :,dds rati,9
z*
?
2EQ
)g1 " )g 21
c #g *
#g
1
1
2
2
2
>)m :1 " )m 9 E n + 1 )m :1 " )m 9 E n ?K
2
1
1
1
2
2
2
8i%ttin%n d Nurmin%n( K = NE:N-19; Farringt,n d 8anning( K =1
)m1 * 1)m2 c )m2 * =; " b " b 2 " Z N1 : x1 + x2 9 :8 E:2 N1 9c b * ":n1 + x2 91 " x1 " n2 c
<
9
)k%wn%ss c,rr%ct%d z n :eart d Nam9c z acc,rding t, Farringt,n d 8anning(
z n * 1 + ZA :A + z 9 " 1 E 2A c V * :1 " )m 9 E:)m n 9 + :1 " )m 9 E:)m n 9 c
>
A * 1 E:RV
z*
2EQ
>
?
1
1 1
2
2 2
9 :1 " )m1 9:1 " 2)m1 9 E: n1)m1 9 2 + :1 " )m2 9:1 " 2)m2 9 E: n2)m2 9 2
?
:)g1 " )m1 9 E:)m1 :1 " )m1 99 " :)g 2 " )m2 9 E:)m2 :1 " )m2 99
1 E:n )m :1 " )m 99 + 1 E: n )m :1 " )m 99 K
1 1
:8i%ttin%n d
Nurmin%n; 1^_a9
1
E R>)m1 :1 " )m1 9:1 " 2)m1 9 E n1 + )m2 :1 " )m2 9:1 " 2)m2 9 E n2 ?
1
2
2
2
8i%ttin%n d Nurmin%n( K=NE:N-19; Farringt,n d 8anning( K =1
)m1 * )m2B E:1 + )m2 :B " 199 c )m2 * =; " b + b 2 + Za: x1 + x2 9 :8 E:2a9c
<
a * n2 :B " 19 c b * n1B + n2 " : x1 + x2 9:B " 19
9
N,t%---xi = succ%ss fr%qu%ncy in gr,up ic ni = sampl% siz% in gr,up ic N * n1 + n2 = t,tal sampl% siz%c )gi * xi E ni c
%C*Cdiff%r%nc% b%tw%%n pr,p,rti,ns p,stulat%d in "0$ 1 * risk rati, p,stulat%d in HOc B * ,dds rati, p,stulat%d in HO
Tabl% ^( T%sts f,r varianc%s
Varianc%s
T%st
T%st
Family
Null
Hyp,th%sis
Eff%ct )iz%
Diff%r%nc% fr,m
c,nstant :,n%
sampl% cas%9
!2 t%sts
#2
Varianc% rati,
c
*1
r*
#2
c
Nth%r
<aram%t%rs
N,nc%ntrality <aram%t%r
$ *O
:H1( c%ntral !2
distributi,n; scal%d with r9
df * N " 1
Pn%quality ,f tw,
varianc%s
F t%sts
# 22
*1
# 12
Varianc% rati,
#2
r * 22
#1
$ *O
:H1( c%ntral F distributi,n;
scal%d with r9
df1 * n1 " 1
df 2 * n2 " 1